solitonssolitons starting in the 19th century, researchers found that certain nonlinear pdes admit...

Chapter 10

Solitons

Starting in the 19th century, researchers found that certain nonlinear PDEs admit exactsolutions in the form of solitary waves, known today as solitons. There’s a famous story ofthe Scottish engineer, John Scott Russell, who in 1834 observed a hump-shaped disturbancepropagating undiminished down a canal. In 1844, he published this observation1, writing,

“I was observing the motion of a boat which was rapidly drawn along anarrow channel by a pair of horses, when the boat suddenly stopped - not so themass of water in the channel which it had put in motion; it accumulated roundthe prow of the vessel in a state of violent agitation, then suddenly leaving itbehind, rolled forward with great velocity, assuming the form of a large solitaryelevation, a rounded, smooth and well-defined heap of water, which continuedits course along the channel apparently without change of form or diminution ofspeed. I followed it on horseback, and overtook it still rolling on at a rate of someeight or nine miles an hour, preserving its original figure some thirty feet longand a foot to a foot and a half in height. Its height gradually diminished, andafter a chase of one or two miles I lost it in the windings of the channel. Such,in the month of August 1834, was my first chance interview with that singularand beautiful phenomenon which I have called the Wave of Translation”.

Russell was so taken with this phenomenon that subsequent to his discovery he built athirty foot wave tank in his garden to reproduce the effect, which was precipitated by aninitial sudden displacement of water. Russell found empirically that the velocity obeyedv ≃

√g(h+ um) , where h is the average depth of the water and um is the maximum

vertical displacement of the wave. He also found that a sufficiently large initial displacementwould generate two solitons, and, remarkably, that solitons can pass through one anotherundisturbed. It was not until 1890 that Korteweg and deVries published a theory of shallowwater waves and obtained a mathematical description of Russell’s soliton.

1J. S. Russell, Report on Waves, 14th Meeting of the British Association for the Advancement of Science,pp. 311-390.

1

2 CHAPTER 10. SOLITONS

Nonlinear PDEs which admit soliton solutions typically contain two important classes ofterms which feed off each other to produce the effect:

DISPERSION −⇀↽− NONLINEARITY

The effect of dispersion is to spread out pulses, while the effect of nonlinearities is, often,to draw in the disturbances. We saw this in the case of front propagation, where dispersionled to spreading and nonlinearity to steepening.

In the 1970’s it was realized that several of these nonlinear PDEs yield entire familiesof exact solutions, and not just isolated solitons. These families contain solutions witharbitrary numbers of solitons of varying speeds and amplitudes, and undergoing mutualcollisions. The three most studied systems have been

• The Korteweg-deVries equation,

ut + 6uux + uxxx = 0 . (10.1)

This is a generic equation for ‘long waves’ in a dispersive, energy-conserving medium,to lowest order in the nonlinearity.

• The Sine-Gordon equation,

φtt − φxx + sinφ = 0 . (10.2)

The name is a play on the Klein-Gordon equation, φtt − φxx + φ = 0. Note that theSine-Gordon equation is periodic under φ→ φ+ 2π.

• The nonlinear Schrödinger equation,

iψt ± ψxx + 2|ψ|2ψ = 0 . (10.3)

Here, ψ is a complex scalar field. Depending on the sign of the second term, we denotethis equation as either NLS(+) or NLS(−), corresponding to the so-called focusing(+) and defocusing (−) cases.

Each of these three systems supports soliton solutions, including exact N -soliton solutions,and nonlinear periodic waves.

10.1 The Korteweg-deVries Equation

Let h0 denote the resting depth of water in a one-dimensional channel, and y(x, t) thevertical displacement of the water’s surface. Let L be a typical horizontal scale of the wave.When |y| ≪ h0, h20 ≪ L2, and v ≈ 0, the evolution of an x-directed wave is described bythe KdV equation,

yt + c0 yx +3c02h0

yyx +16c0 h

20 yxxx = 0 , (10.4)

10.1. THE KORTEWEG-DEVRIES EQUATION 3

where c0 =√gh0. For small amplitude disturbances, only the first two terms are conse-

quential, and we haveyt + c0 yx ≈ 0 , (10.5)

the solution to which isy(x, t) = f(x− c0t) , (10.6)

where f(ξ) is an arbitrary shape; the disturbance propagates with velocity c0. When thedispersion and nonlinearity are included, only a particular pulse shape can propagate in anundistorted manner; this is the soliton.

It is convenient to shift to a moving frame of reference:

x̃ = x− c0t , t̃ = t , (10.7)

hence∂

∂x=

∂

∂x̃,

∂

∂t=

∂

∂t̃− c0

∂

∂x̃. (10.8)

Thus,

yet+ c0 yex +

3c02h0

y yex +

16c0 h

20 yexexex = 0 . (10.9)

Finally, rescaling position, time, and displacement, we arrive at the KdV equation ,

ut + 6uux + uxxx = 0 , (10.10)

which is a convenient form.

10.1.1 KdV solitons

We seek a solution to the KdV equation of the form u(x, t) = u(x − V t). Then withξ ≡ x− V t, we have ∂x = ∂ξ and ∂t = −V ∂ξ when acting on u(x, t) = u(ξ). Thus, we have

−V u′ + 6uu′ + u′′′ = 0 . (10.11)

Integrating once, we have−V u+ 3u2 + u′′ = A , (10.12)

where A is a constant. We can integrate once more, obtaining

−12V u2 + u3 + 12(u

′)2 = Au+B , (10.13)

where now both A and B are constants. We assume that u and all its derivatives vanish inthe limit ξ → ±∞, which entails A = B = 0. Thus,

du

dξ= ±u

√V − 2u . (10.14)

With the substitutionu = 12V sech

2θ , (10.15)


Figure 10.1: Soliton solutions to the KdV equation, with five evenly spaced V values rangingfrom V = 2 (blue) to V = 10 (orange). The greater the speed, the narrower the shape.

we find dθ = ∓12√V dξ, hence we have the solution

u(x, t) = 12V sech2(√

V2 (x− V t− ξ0)

). (10.16)

Note that the maximum amplitude of the soliton is umax =12V , which is proportional to

its velocity V . The KdV equation imposes no limitations on V other than V ≥ 0.

10.1.2 Periodic solutions : soliton trains

If we relax the condition A = B = 0, new solutions to the KdV equation arise. Define thecubic

P (u) = 2u3 − V u2 − 2Au− 2B (10.17)≡ 2(u− u1)(u− u2)(u− u3) ,

where ui = ui(A,B, V ). We presume that A, B, and V are such that all three rootsu1,2,3 are real and nondegenerate. Without further loss of generality, we may then assumeu1 < u2 < u3. Then

du

dξ= ±

√−P (u) . (10.18)

Since P (u) < 0 for u2 < u < u3, we conclude u(ξ) must lie within this range. Therefore,we have

ξ − ξ0 = ±u∫

u2

ds√−P (s)

(10.19)

= ±(

2

u3 − u1

)1/2 φ∫

0

dθ√1 − k2 sin2θ

,


Figure 10.2: The Jacobi elliptic functions sn(ζ, k) (solid) and cn(ζ, k) (dot-dash) versusζ/K(k), for k = 0 (blue), k = 1√

2(green), and k = 0.9 (red).

where

u ≡ u3 − (u3 − u2) sin2φ (10.20)

k2 ≡ u3 − u2u3 − u1

. (10.21)

The solution for u(ξ) is then

u(ξ) = u3 − (u3 − u2) sn2(ζ, k) , (10.22)

where

ζ =

√u3 − u1

2

(ξ − ξ0

)(10.23)

and sn(ζ, k) is the Jacobi elliptic function.

10.1.3 Interlude: primer on elliptic functions

We assume 0 ≤ k2 ≤ 1 and we define

ζ(φ, k) =

φ∫

0


. (10.24)

The sn and cn functions are defined by the relations

sn(ζ, k) = sinφ (10.25)

cn(ζ, k) = cosφ . (10.26)


Figure 10.3: The cubic function P (u) (left), and the soliton lattice (right) for the caseu1 = −1.5, u2 = −0.5, and u3 = 2.5.

Note that sn2(ζ, k) + cn2(ζ, k) = 1. One also defines the function dn(ζ, k) from the relation

dn2(ζ, k) + k2 sn2(ζ, k) = 1 . (10.27)

When φ advances by one period, we have ∆φ = 2π, and therefore ∆ζ = Z, where

Z =

2π∫

0


= 4 K(k) , (10.28)

where K(k) is the complete elliptic integral of the first kind. Thus, sn(ζ + Z, k) = sn(ζ, k),and similarly for the cn function. In fig. 10.2, we sketch the behavior of the ellipticfunctions over one quarter of a period. Note that for k = 0 we have sn(ζ, 0) = sin ζ andcn(ζ, 0) = cos ζ.

10.1.4 The soliton lattice

Getting back to our solution in eqn. 10.22, we see that the solution describes a solitonlattice with a wavelength

λ =

√8 K(k)√u3 − u1

. (10.29)

Note that our definition of P (u) entails

V = 2(u1 + u2 + u3) . (10.30)

There is a simple mechanical analogy which merits illumination. Suppose we define

W (u) ≡ u3 − 12V u2 −Au , (10.31)


Figure 10.4: The cubic function P (u) (left), and the soliton lattice (right) for the caseu1 = −1.5, u2 = −1.49, and u3 = 2.50.

and furthermore E ≡ B. Then

1

2

(du

dξ

)2+W (u) = E , (10.32)

which takes the form of a one-dimensional Newtonian mechanical system, if we replaceξ → t and interpret uξ as a velocity. The potential is W (u) and the total energy is E. Interms of the polynomial P (u), we have P = 2(W − E). Accordingly, the ‘motion’ u(ξ) isflattest for the lowest values of u, near u = u2, which is closest to the local maximum ofW (u).

Note that specifying umin = u2, umax = u3, and the velocity V specifies all the parameters.Thus, we have a three parameter family of soliton lattice solutions.

10.1.5 N-soliton solutions to KdV

In 1971, Ryogo Hirota2 showed that exact N -soliton solutions to the KdV equation exist.Here we discuss the Hirota solution, following the discussion in the book by Whitham.

The KdV equation may be written as

ut +{3u2 + uxx

}x

= 0 , (10.33)

which is in the form of the one-dimensional continuity equation ut+jx = 0, where the currentis j = 3u2 + uxx. Let us define u = px. Then our continuity equation reads ptx + jx = 0,which can be integrated to yield pt + j = C, where C is a constant. Demanding that u and

2R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).


its derivatives vanish at spatial infinity requires C = 0. Hence, we have

pt + 3p2x + pxxx = 0 . (10.34)

Now consider the nonlinear transformation

p = 2 (lnF )x =2FxF

. (10.35)

We then have

pt =2FxtF

− 2FxFtF 2

(10.36)

px =2FxxF

− 2F2x

F 2(10.37)

and

pxx =2FxxxF

− 6FxFxxF 2

+4F 3xF 3

(10.38)

pxxx =2FxxxxF

− 8FxFxxxF 2

− 6Fxx2

F 2+

24F 2xFxxF 3

− 12F4x

F 4. (10.39)

When we add up the combination pt + 3p2x + pxxx = 0, we find, remarkably, that the terms

with F 3 and F 4 in the denominator cancel. We are then left with

F(Ft + Fxxx

)x− Fx

(Ft + Fxxx

)+ 3(F 2xx − FxFxxx

)= 0 . (10.40)

This equation has the two-parameter family of solutions

F (x, t) = 1 + eφ(x,t) . (10.41)

whereφ(x, t) = α (x− b− α2 t) , (10.42)

with α and b constants. Note that these solutions are all annihilated by the operator ∂t+∂3x,

and also by the last term in eqn. 10.40 because of the homogeneity of the derivatives.Converting back to our original field variable u(x, t), we have that these solutions are singlesolitons:

u = px =2(FFxx − F 2x )

F 2=

α2 f

(1 + f)2= 12α

2 sech2(12φ) . (10.43)

The velocity for these solutions is V = α2.

If eqn. 10.40 were linear, our job would be done, and we could superpose solutions. Wewill meet up with such a felicitous situation when we discuss the Cole-Hopf transformationfor the one-dimensional Burgers’ equation. But for KdV the situation is significantly moredifficult. We will write

F = 1 + F (1) + F (2) + . . .+ F (N) , (10.44)


withF (1) = f1 + f2 + . . .+ fN , (10.45)

where

fj(x, t) = eφj(x,t) (10.46)

φj(x, t) = αj (x− α2j t− bj) . (10.47)

We may then derive a hierarchy of equations, the first two levels of which are

(F

(1)t + F

(1)xxx

)x

= 0 (10.48)(F

(2)t + F

(2)xxx

)x

= −3(F (1)xx F

(1)xx − F (1)x F (1)xxx

). (10.49)

Let’s explore the case N = 2. The equation for F (2) becomes

(F

(2)t + F

(2)xxx

)x

= 3α1α2 (α2 − α1)2 f1f2 , (10.50)

with solution

F (2) =

(α1 − α2α1 + α2

)2f1f2 . (10.51)

Remarkably, this completes the hierarchy for N = 2. Thus,

F = 1 + f1 + f2 +

(α1 − α2α1 + α2

)2f1f2 (10.52)

= det

1 + f1

2√

α1α

2

α1+α

2

f12√

α1α

2

α1+α

2

f2 1 + f2

.

What Hirota showed, quite amazingly, is that this result generalizes to the N -soliton case,

F = det Q , (10.53)

where Q is the symmetric matrix,

Qmn = δmn +2√αmαn

αm + αnfm fn . (10.54)

Thus, N -soliton solutions to the KdV equation may be written in the form

u(x, t) = 2∂2

∂x2ln detQ(x, t) . (10.55)

Consider the case N = 2. Direct, if tedious, calculations lead to the expression

u = 2α21f1 + α

22f2 + 2 (α1 − α2)2f1f2 +

(α1−α2α

1+α

2

)2(α21f1f

22 + α

22f

21 f2)

[1 + f1 + f2 +

(α1−α

2

α1+α

2

)2f1f2

]2 . (10.56)


Figure 10.5: Early and late time configuration of the two soliton solution to the KdVequation.

Recall that

fj(x, t) = exp[αj (xj − α2j t− bj)

]. (10.57)

Let’s consider (x, t) values for which f1 ≃ 1 is neither large nor small, and investigate whathappens in the limits f2 ≪ 1 and f2 ≫ 1. In the former case, we find

u ≃ 2α21 f1

(1 + f1)2

(f2 ≪ 1) , (10.58)

which is identical to the single soliton case of eqn. 10.43. In the opposite limit, we have

u ≃ 2α21 g1

(1 + g1)2

(f2 ≫ 1) , (10.59)

where

g1 =

(α1 − α2α1 + α2

)2f1 (10.60)

But multiplication of fj by a constant C is equivalent to a translation:

C fj(x, t) = fj(x+ α−1j lnC , t) ≡ fj(x− ∆xj , t) . (10.61)

Thus, depending on whether f2 is large or small, the solution either acquires or does notacquire a spatial shift ∆x1, where

∆xj =2

αjln

∣∣∣∣α1 + α2α1 − α2

∣∣∣∣ . (10.62)


Figure 10.6: Spacetime diagram for the collision of two KdV solitons. The strong, fastsoliton (#1) is shifted forward and the weak slow one (#2) is shifted backward. Thered and blue lines indicate the centers of the two solitons. The yellow shaded circle isthe ‘interaction region’ where the solution is not simply a sum of the two single solitonwaveforms.

The function f(x, t) = exp[α(x − α2t − b)

]is monotonically increasing in x (assuming

α > 0). Thus, if at fixed t the spatial coordinate x is such that f ≪ 1, this means that thesoliton lies to the right. Conversely, if f ≫ 1 the soliton lies to the left. Suppose α1 > α2,in which case soliton #1 is stronger (i.e. greater amplitude) and faster than soliton #2.The situation is as depicted in figs. 10.5 and 10.6. Starting at early times, the strongsoliton lies to the left of the weak soliton. It moves faster, hence it eventually overtakes theweak soliton. As the strong soliton passes through the weak one, it is shifted forward , andthe weak soliton is shifted backward . It hardly seems fair that the strong fast soliton getspushed even further ahead at the expense of the weak slow one, but sometimes life is justlike that.

10.1.6 Bäcklund transformations

For certain nonlinear PDEs, a given solution may be used as a ‘seed’ to generate an entirehierarchy of solutions. This is familiar from the case of Riccati equations, which are non-linear and nonautonomous ODEs, but for PDEs it is even more special. The general formof the Bäcklund transformation (BT) is

u1,t = P (u1 , u0 , u0,t , u0,x) (10.63)

u1,x = Q(u1 , u0 , u0,t , u0,x) , (10.64)

where u0(x, t) is the known solution.


Figure 10.7: “Wrestling’s living legend” Bob Backlund, in a 1983 match, is subjected to adevastating camel clutch by the Iron Sheikh. The American Bob Backlund has nothing todo with the Bäcklund transformations discussed in the text, which are named for the 19th

century Swedish mathematician Albert Bäcklund. Note that Bob Backlund’s manager hasthrown in the towel (lower right).

A Bäcklund transformation for the KdV equation was first obtained in 19733. This provideda better understanding of the Hirota hierarchy. To proceed, following the discussion in thebook by Scott, we consider the earlier (1968) result of Miura4, who showed that if v(x, t)satisfies the modified KdV (MKdV) equation5,

vt − 6v2vx + vxxx = 0 , (10.65)

thenu = −(v2 + vx) (10.66)

solves KdV:

ut + 6uux + uxxx = −(v2 + vx)t + 6(v2 + vx)(v2 + vx)x − (v2 + vx)xxx= −

(2v + ∂x

)(vt − 6v2vx + vxxx

)= 0 . (10.67)

From this result, we have that if

vt − 6(v2 + λ) vx + vxxx = 0 , (10.68)

thenu = −(v2 + vx + λ) (10.69)

solves KdV. The MKdV equation, however, is symmetric under v → −v, hence

u0 = −vx − v2 − λ (10.70)u1 = +vx − v2 − λ (10.71)

3H. D. Wahlquist and F. B. Eastabrook, Phys. Rev. Lett. 31, 1386 (1973).4R. M. Miura, J. Math. Phys. 9, 1202 (1968).5Note that the second term in the MKdV equation is proportional to v2vx, as opposed to uux which

appears in KdV.


both solve KdV. Now define u0 ≡ −w0,x and u1 ≡ −w1,x. Subtracting the above twoequations, we find

u0 − u1 = −2vx ⇒ w0 − w1 = 2v . (10.72)Adding the equations instead gives

w0,x + w1,x = 2(v2 + λ)

= 12(w0 − w1)2 + 2λ . (10.73)

Substituting for v = 12(w0 − w1) and v2 + λ− 12(w0,x + w1,x) into the MKdV equation, wehave

(w0 − w1)t − 3(w20,x − w21,x

)+ (w0 − w1)xxx = 0 . (10.74)

This last equation is a Bäcklund transformation (BT), although in a somewhat nonstandardform, since the RHS of eqn. 10.63, for example, involves only the ‘new’ solution u1 andthe ‘old’ solution u0 and its first derivatives. Our equation here involves third derivatives.

However, we can use eqn. 10.73 to express w1,x in terms of w0, w0,x, and w1.

Starting from the trivial solution w0 = 0, eqn. 10.73 gives

w1,x =12w

21 + λ . (10.75)

With the choice λ = −14α2 < 0, we integrate and obtain

w1(x, t) = −2α tanh(

12αx+ ϕ(t)

), (10.76)

were ϕ(t) is at this point arbitrary. We fix ϕ(t) by invoking eqn. 10.74, which says

w1,t = 3w21,x − w1,xxx = 0 . (10.77)

Invoking w1,x =12w

21 +λ and differentiating twice to obtain w1,xxx, we obtain an expression

for the RHS of the above equation. The result is w1,t + α2w1,x = 0, hence

w1(x, t) = −α tanh[

12α (x− α

2t− b)]

(10.78)

u1(x, t) =12α

2 sech2[

12α (x− α2t− b)

], (10.79)

which recapitulates our earlier result. Of course we would like to do better, so let’s try toinsert this solution into the BT and the turn the crank and see what comes out. This isunfortunately a rather difficult procedure. It becomes tractable if we assume that successiveBäcklund transformations commute, which is the case, but which we certainly have not yetproven. That is, we assume that w12 = w21, where

w0λ1−−−−→ w1

λ2−−−−→ w12 (10.80)

w0λ2−−−−→ w2

λ1−−−−→ w21 . (10.81)

Invoking this result, we find that the Bäcklund transformation gives

w12 = w21 = w0 −4(λ1 − λ2)w1 − w2

. (10.82)

Successive applications of the BT yield Hirota’s multiple soliton solutions:

w0λ1−−−−→ w1

λ2−−−−→ w12

λ3−−−−→ w123

λ4−−−−→ · · · . (10.83)


10.2 Sine-Gordon Model

Consider transverse electromagnetic waves propagating down a superconducting transmis-sion line, shown in fig. 10.8. The transmission line is modeled by a set of inductors,capacitors, and Josephson junctions such that for a length dx of the transmission line,the capacitance is dC = C dx, the inductance is dL = L dx, and the critical current isdI0 = I0 dx. Dividing the differential voltage drop dV and shunt current dI by dx, weobtain

∂V

∂x= −L ∂I

∂t(10.84)

∂I

∂x= −C ∂V

∂t− I0 sinφ , (10.85)

where φ is the difference φ = φupper − φlower in the superconducting phases. The voltage isrelated to the rate of change of φ through the Josephson equation,

∂φ

∂t=

2eV

~, (10.86)

and therefore∂φ

∂x= −2eL

~I . (10.87)

Thus, we arrive at the equation

1

c2∂2φ

∂t2− ∂

2φ

∂x2+

1

λ2Jsinφ = 0 , (10.88)

where c = (LC)−1/2 is the Swihart velocity and λJ = (~/2eLI0)1/2 is the Josephson length.We may now rescale lengths by λJ and times by λJ/c to arrive at the sine-Gordon equation,

φtt − φxx + sinφ = 0 . (10.89)

This equation of motion may be derived from the Lagrangian density

L = 12φ2t − 12φ

2x − U(φ) . (10.90)

We then obtain

φtt − φxx = −∂U

∂φ, (10.91)

and the sine-Gordon equation follows for U(φ) = 1 − cosφ.

Assuming φ(x, t) = φ(x − V t) we arrive at (1 − V 2)φξξ = U ′(ξ), and integrating once weobtain

12 (1 − V

2)φ2ξ − U(φ) = E . (10.92)This describes a particle of mass M = 1 − V 2 moving in the inverted potential −U(φ).Assuming V 2 < 1, we may solve for φξ:

φ2ξ =2(E + U(φ))

1 − V 2 , (10.93)

10.2. SINE-GORDON MODEL 15

Figure 10.8: A superconducting transmission line is described by a capacitance per unitlength C, an inductance per unit length L, and a critical current per unit length I0. Basedon fig. 3.4 of A. Scott, Nonlinear Science.

which requires E ≥ −Umax in order for a solution to exist. For −Umax < E < −Umin,the motion is bounded by turning points. The situation for the sine-Gordon (SG) model issketched in fig. 10.9. For the SG model, the turning points are at φ∗± = ± cos−1(E + 1),with −2 < E < 0. We can write

φ∗± = π ± δ , δ = 2cos−1√

−E2. (10.94)

This class of solutions describes periodic waves. From

√2 dξ√

1 − V 2=

dφ√E + U(φ)

, (10.95)

we have that the spatial period λ is given by

λ =√

2 (1 − V 2)φ∗+∫

φ∗−

dφ√E + U(φ)

. (10.96)

If E > −Umin, then φξ is always of the same sign, and φ(ξ) is a monotonic function of ξ.If U(φ) = U(φ + 2π) is periodic, then the solution is a ‘soliton lattice’ where the spatialperiod of φ mod 2π is

λ̃ =

√1 − V 2

2

2π∫

0

dφ√E + U(φ)

. (10.97)


Figure 10.9: The inverted potential −U(φ) = cosφ− 1 for the sine-Gordon problem.

For the sine-Gordon model, with U(φ) = 1 − cosφ, one finds

λ =

√−π (1 − V

2)

E(E + 2), λ̃ =

√8 (1 − V 2)E + 2

K

(√2

E + 2

). (10.98)

10.2.1 Tachyon solutions

When V 2 > 1, we have12(V

2 − 1)φ2ξ + U(φ) = −E . (10.99)

Such solutions are called tachyonic. There are again three possibilities:

• E > Umin : no solution.

• −Umax < E < −Umin : periodic φ(ξ) with oscillations about φ = 0.

• E < −Umax : tachyon lattice with monotonic φ(ξ).

It turns out that the tachyon solution is unstable.

10.2.2 Hamiltonian formulation

The Hamiltonian density is

H = π φt − L , (10.100)

where

π =∂L∂φt

= φt (10.101)


is the momentum density conjugate to the field φ. Then

H(π, φ) = 12π2 + 12φ

2x + U(φ) . (10.102)

Note that the total momentum in the field is

P =

∞∫

−∞

dx π =

∞∫

−∞

dx φt = −V∞∫

−∞

dx φx

= −V[φ(∞) − φ(−∞)

]= −2πnV , (10.103)

where n =[φ(∞) − φ(−∞)

]/2π is the winding number .

10.2.3 Phonons

The Hamiltonian density for the SG system is minimized when U(φ) = 0 everywhere. Theground states are then classified by an integer n ∈ Z, where φ(x, t) = 2πn for ground staten. Suppose we linearize the SG equation about one of these ground states, writing

φ(x, t) = 2πn+ η(x, t) , (10.104)

and retaining only the first order term in η from the nonlinearity. The result is the Klein-Gordon (KG) equation,

ηtt − ηxx + η = 0 . (10.105)This is a linear equation, whose solutions may then be superposed. Fourier transformingfrom (x, t) to (k, ω), we obtain the equation

(− ω2 + k2 + 1

)η̂(k, ω) = 0 , (10.106)

which entails the dispersion relation ω = ±ω(k), where

ω(k) =√

1 + k2 . (10.107)

Thus, the most general solution to the (1 + 1)-dimensional KG equation is

η(x, t) =

∞∫

−∞

dk

2π

{A(k) eikx e−i

√1+k2 t +B(k) eikx ei

√1+k2 t

}. (10.108)

For the Helmholtz equation ηtt−ηxx = 0, the dispersion is ω(k) = |k|, and the solution maybe written as η(x, t) = f(x− t) + g(x + t), which is the sum of arbitrary right-moving andleft-moving components. The fact that the Helmholtz equation preserves the shape of thewave is a consequence of the absence of dispersion, i.e. the phase velocity vp(k) =

ωk is the

same as the group velocity vg(k) =∂ω∂k . This is not the case for the KG equation, obviously,

since

vp(k) =ω

k=

√1 + k2

k, vg(k) =

∂ω

∂k=

k√1 + k2

, (10.109)

hence vpvg = 1 for KG.


10.2.4 Mechanical realization

The sine-Gordon model can be realized mechanically by a set of pendula elastically coupled.The kinetic energy T and potential energy U are given by

T =∑

n

12mℓ

2φ̇2n (10.110)

U =∑

n

[12κ(φn+1 − φn

)2+mgℓ

(1 − cosφn

)]. (10.111)

Here ℓ is the distance from the hinge to the center-of-mass of the pendulum, and κ is thetorsional coupling. From the Euler-Lagrange equations we obtain

mℓ2φ̈n = −κ(φn+1 + φn−1 − 2φn

)−mgℓ sinφn . (10.112)

Let a be the horizontal spacing between the pendula. Then we can write the above equationas

φ̈n =

≡ c2︷︸︸︷κa2

mℓ2·

≈φ′′n︷︸︸︷1

a

[(φn+1 − φn

a

)−(φn − φn−1

a

)]−gℓ

sinφn . (10.113)

The continuum limit of these coupled ODEs yields the PDE

1

c2φtt − φxx +

1

λ2sinφ = 0 , (10.114)

which is the sine-Gordon equation, with λ = (κa2/mgℓ)1/2.

10.2.5 Kinks and antikinks

Let us return to eqn. 10.92 and this time set E = −Umin. With U(φ) = 1 − cosφ, we haveUmin = 0, and thus

dφ

dξ= ± 2√

1 − V 2sin(

12φ). (10.115)

This equation may be integrated:

± dξ√1 − V 2

=dφ

2 sin 12φ= d ln tan 14φ . (10.116)

Thus, the solution is

φ(x, t) = 4 tan−1 exp

(± x− V t− x0√

1 − V 2

). (10.117)

where ξ0 is a constant of integration. This describes either a kink (with dφ/dx > 0) oran antikink (with dφ/dx < 0) propagating with velocity V , instantaneously centered atx = x0 + V t. Unlike the KdV soliton, the amplitude of the SG soliton is independent of its


Figure 10.10: Kink and antikink solutions to the sine-Gordon equation.

velocity. The SG soliton is topological , interpolating between two symmetry-related vacuumstates, namely φ = 0 and φ = 2π.

Note that the width of the kink and antikink solutions decreases as V increases. This isa Lorentz contraction, and should have been expected since the SG equation possesses aLorentz invariance under transformations

x =x′ + vt′√

1 − v2(10.118)

t =t′ + vx′√

1 − v2. (10.119)

One then readily finds∂2t − ∂2x = ∂2t′ − ∂2x′ . (10.120)

The moving soliton solutions may then be obtained by a Lorentz transformation of thestationary solution,

φ(x, t) = 4 tan−1 e±(x−x0) . (10.121)

The field φ itself is a Lorentz scalar, and hence does not change in magnitude under aLorentz transformation.

10.2.6 Bäcklund transformation for the sine-Gordon system

Recall D’Alembert’s method of solving the Helmholtz equation, by switching to variables

ζ = 12(x− t) ∂x = 12∂ζ + 12∂t (10.122)τ = 12(x+ t) ∂t = −12∂ζ + 12∂t . (10.123)

The D’Alembertian operator then becomes

∂2t − ∂2x = − ∂ζ ∂τ . (10.124)This transforms the Helmholtz equation φtt − φxx = 0 to φζτ = 0, with solutions φ(ζ, τ) =f(ζ) + g(τ), with f and g arbitrary functions of their arguments. As applied to the SGequation, we have

φζτ = sinφ . (10.125)


Suppose we have a solution φ0 to this equation. Suppose further that φ1 satisfies the pairof equations,

φ1,ζ = 2λ sin

(φ1 + φ0

2

)+ φ0,ζ (10.126)

φ1,τ =2

λsin

(φ1 − φ0

2

)− φ0,τ . (10.127)

Thus,

φ1,ζτ − φ0,ζτ = λ cos(φ1 + φ0

2

)(φ1,τ − φ0,τ

)

= 2cos

(φ1 + φ0

2

)sin

(φ1 − φ0

2

)

= sinφ1 − sinφ0 . (10.128)

Thus, if φ0,ζτ = sinφ0, then φ1,ζτ = sinφ1 as well, and φ1 satisfies the SG equation. Eqns.10.126 and 10.127 constitute a Bäcklund transformation for the SG system.

Let’s give the ‘Bäcklund crank’ one turn, starting with the trivial solution φ0 = 0. We thenhave

φ1,ζ = 2λ sin12φ1 (10.129)

φ1,τ = 2λ−1 sin 12φ1 . (10.130)

The solution to these equations is easily found by direct integration:

φ(ζ, τ) = 4 tan−1 eλζ eτ/λ . (10.131)

In terms of our original independent variables (x, t), we have

λζ + λ−1τ = 12(λ+ λ−1

)x− 12

(λ− λ−1

)t = ± x− vt√

1 − v2, (10.132)

where

v ≡ λ2 − 1λ2 + 1

⇐⇒ λ = ±(

1 + v

1 − v

)1/2. (10.133)

Thus, we generate the kink/antikink solution

φ1(x, t) = 4 tan−1 exp

(± x− vt√

1 − v2

). (10.134)

As was the case with the KdV system, successive Bäcklund transformations commute. Thus,

φ0λ1−−−−→ φ1

λ2−−−−→ φ12 (10.135)φ0

λ2−−−−→ φ2

λ1−−−−→ φ21 , (10.136)


with φ12 = φ21. This allows one to eliminate the derivatives and write

tan

(φ12 − φ0

4

)=

(λ1 + λ2λ1 − λ2

)tan

(φ2 − φ1

4

). (10.137)

We can now create new solutions from individual kink pairs (KK), or kink-antikink pairs

(KK). For the KK system, taking v1 = v2 = v yields

φKK(x, t) = 4 tan−1(v sinh(γx)

cosh(γvt)

), (10.138)

where γ is the Lorentz factor,

γ =1√

1 − v2. (10.139)

Note that φKK(±∞, t) = ±2π and φKK(0, t) = 0, so there is a phase increase of 2π on eachof the negative and positive half-lines for x, and an overall phase change of +4π. For theKK system, if we take v1 = −v2 = v, we obtain the solution

φKK

(x, t) = 4 tan−1(

sinh(γvt)

v cosh(γx)

). (10.140)

In this case, analytically continuing to imaginary v with

v =iω√

1 − ω2=⇒ γ =

√1 − ω2 (10.141)

yields the stationary breather solution,

φB(x, t) = 4 tan−1

( √1 − ω2 sin(ωt)

ω cosh(√

1 − ω2 x)

). (10.142)

The breather is a localized solution to the SG system which oscillates in time. By applyinga Lorentz transformation of the spacetime coordinates, one can generate a moving breathersolution as well.

Please note, in contrast to the individual kink/antikink solutions, the solutions φKK, φKK,

and φB are not functions of a single variable ξ = x−V t. Indeed, a given multisoliton solutionmay involve components moving at several different velocities. Therefore the total momen-tum P in the field is no longer given by the simple expression P = V

(φ(−∞) − φ(+∞)

).

However, in cases where the multikink solutions evolve into well-separated solitions, as hap-pens when the individual kink velocities are distinct, the situation simplifies, as we mayconsider each isolated soliton as linearly independent. We then have

P = −2π∑

i

ni Vi , (10.143)

where ni = +1 for kinks and ni = −1 for antikinks.


10.3 Nonlinear Schrödinger Equation

The Nonlinear Schrödinger (NLS) equation arises in several physical contexts. One is theGross-Pitaevskii action for an interacting bosonic field,

S[ψ,ψ∗] =

∫dt

∫ddx

{iψ∗

∂ψ

∂t− ~

2

2m∇ψ∗ ·∇ψ − U

(ψ∗ψ

)+ µψ∗ψ

}, (10.144)

where ψ(x, t) is a complex scalar field. The local interaction U(|ψ|2

)is taken to be quartic,

U(|ψ|2

)= 12g |ψ|

4 . (10.145)

Note that

U(|ψ|2

)− µ |ψ|2 = 12g

(|ψ|2 − µ

g

)2− µ

2

2g. (10.146)

Extremizing the action with respect to ψ∗ yields the equation

δS

δψ∗= 0 = i

∂ψ

∂t+

~2

2m∇

2ψ − U ′(ψ∗ψ

)ψ + µψ . (10.147)

Extremization with respect to ψ yields the complex conjugate equation. In d = 1, we have

iψt = −~

2

2mψxx + U

′(ψ∗ψ)ψ − µψ . (10.148)

We can absorb the chemical potential by making a time-dependent gauge transformation

ψ(x, t) −→ eiµt ψ(x, t) . (10.149)

Further rescalings of the field and independent variables yield the generic form

iψt ± ψxx + 2 |ψ|2 ψ = 0 , (10.150)

where the + sign pertains for the case g < 0 (attactive interaction), and the − sign forthe case g > 0 (repulsive interaction). These cases are known as focusing , or NLS(+), anddefocusing , or NLS(−), respectively.

10.3.1 Amplitude-phase representation

We can decompose the complex scalar ψ into its amplitude and phase:

ψ = Aeiφ . (10.151)

We then find

ψt =(At + iAφt

)eiφ (10.152)

ψx =(Ax + iAφx

)eiφ (10.153)

ψxx =(Axx −Aφ2x + 2iAxφx + iAφxx

)eiφ . (10.154)

10.3. NONLINEAR SCHRÖDINGER EQUATION 23

Multiplying the NLS(±) equations by e−iφ and taking real and imaginary parts, we obtainthe coupled nonlinear PDEs,

−Aφt ±(Axx −Aφ2x

)+ 2A3 = 0 (10.155)

At ±(2Axφx +Aφxx

)= 0 . (10.156)

Note that the second of these equations may be written in the form of a continuity equation,

ρt + jx = 0 , (10.157)

where

ρ = A2 (10.158)

j = ± 2A2φx . (10.159)

10.3.2 Phonons

One class of solutions to NLS(±) is the spatially uniform case

ψ0(x, t) = A0 e2iA20 t , (10.160)

with A = A0 and φ = 2A20 t. Let’s linearize about these solutions, writing

A(x, t) = A0 + δA(x, t) (10.161)

φ(x, t) = 2A20 t+ δφ(x, t) . (10.162)

Our coupled PDEs then yield

4A20 δA± δAxx −A0 δφt = 0 (10.163)δAt ±A0 δφxx = 0 . (10.164)

Fourier transforming, we obtain

(4A20 ∓ k2 iA0 ω

−iω ∓A0k2

)(δÂ(k, ω)

δφ̂(k, ω)

)= 0 . (10.165)

Setting the determinant to zero, we obtain

ω2 = ∓4A20 k2 + k4 . (10.166)

For NLS(−), we see that ω2 ≥ 0 for all k, meaning that the initial solution ψ0(x, t) is stable.For NLS(+), however, we see that wavevectors k ∈

[−2A0 , 2A0

]are unstable. This is

known as the Benjamin-Feir instability .


10.3.3 Soliton solutions for NLS(+)

Let’s consider moving soliton solutions for NLS(+). We try a two-parameter solution of theform

A(x, t) = A(x− ut) (10.167)φ(x, t) = φ(x− vt) . (10.168)

This results in the coupled ODEs

Axx −Aφ2x + vAφx + 2A3 = 0 (10.169)Aφxx + 2Axφx − uAx = 0 . (10.170)

Multiplying the second equation by 2A yields

((2φx − u

)A2)x

= 0 =⇒ φx = 12u+P

2A2, (10.171)

where P is a constant of integration. Inserting this in the first equation results in

Axx + 2A3 + 14(2uv − u

2)A+ 12(v − u)PA−1 − 14PA

−3 = 0 . (10.172)

We may write this asAxx +W

′(A) = 0 , (10.173)

whereW (A) = 12A

4 + 18 (2uv − u2)A2 + 12(v − u)P lnA+ 18PA

−2 (10.174)

plays the role of a potential. We can integrate eqn. 10.173 to yield

12A

2x +W (A) = E , (10.175)

where E is second constant of integration. This may be analyzed as a one-dimensionalmechanics problem.

The simplest case to consider is P = 0, in which case

W (A) = 12(A2 + 12uv − 14u

2)A2 . (10.176)

If u2 < 2uv, then W (A) is everywhere nonnegative and convex, with a single global min-imum at A = 0, where W (0) = 0. The analog mechanics problem tells us that A willoscillate between A = 0 and A = A∗, where W (A∗) = E > 0. There are no solutions withE < 0. If u2 > 2uv, then W (A) has a double well shape6. If E > 0 then the oscillationsare still between A = 0 and A = A∗, but if E < 0 then there are two positive solutions toW (A) = E. In this latter case, we may write

F (A) ≡ 2[E −W (A)

]=(A2 −A20

)(A21 −A2

), (10.177)

6Although we have considered A > 0 to be an amplitude, there is nothing wrong with allowing A < 0.When A(t) crosses A = 0, the phase φ(t) jumps by ± π.

10.3. NONLINEAR SCHRÖDINGER EQUATION 25

where A0 < A1 and

E = −12A20A

21 (10.178)

14u

2 − 12uv = A20 +A

21 . (10.179)

The amplitude oscillations are now between A = A∗0 and A = A∗1. The solution is given in

terms of Jacobi elliptic functions:

ψ(x, t) = A1 exp[

i2u (x− vt)

]dn(A1(x− ut − ξ0) , k

), (10.180)

where

k2 = 1 − A20

A21. (10.181)

The simplest case is E = 0, for which A0 = 0. We then obtain

ψ(x, t) = A∗ exp[

i2u (x− vt)

]sech

(A∗(x− ut− ξ0)

), (10.182)

where 4A∗2 = u2 − 2uv. When v = 0 we obtain the stationary breather solution, for whichthe entire function ψ(x, t) oscillates uniformly.

10.3.4 Dark solitons for NLS(−)

The small oscillations of NLS(−) are stable, as we found in our phonon calculation. It istherefore perhaps surprising to note that this system also supports solitons. We write

ψ(x, t) =√ρ0 e

2iρ0t Z(x, t) , (10.183)

which leads to

iZt = Zxx + 2ρ0(1 − |Z|2

)Z . (10.184)

We then write Z = X + iY , yielding

Xt = Yxx + 2ρ0(1 −X2 − Y 2

)Y (10.185)

−Yt = Xxx + 2ρ0(1 −X2 − Y 2

)X . (10.186)

We try Y = Y0, a constant, and set X(x, t) = X(x− V t). Then

−V Xξ = 2ρ0Y0(1 − Y 20 −X2

)(10.187)

0 = Xξξ + 2ρ0(1 − Y 20 −X2

)X (10.188)

Thus,

Xξ = −2ρ0Y0V

(1 − Y 20 −X2

)(10.189)


from which it follows that

Xξξ =4ρ0Y0V

XXξ

= −8ρ20Y

20

V

(1 − Y 20 −X2

)X =

4ρ0Y20

V 2Xξξ . (10.190)

Thus, in order to have a solution, we must have

V = ±2√ρ0 Y0 . (10.191)

We now write ξ = x− V t, in which case

±√ρ0 dξ =

dX

1 − Y 20 −X2. (10.192)

From this point, the derivation is elementary. One writes X =√

1 − Y 20 X̃ , and integratesto obtain

X̃(ξ) = ∓ tanh(√

1 − Y 20√ρ0 (ξ − ξ0)

). (10.193)

We simplify the notation by writing Y0 = sin β. Then

ψ(x, t) =√ρ0 e

iα e2iρ0t[

cosβ tanh(√

ρ0 cos(β(x− V t− ξ0

))− i sin β

], (10.194)

where α is a constant. The density ρ = |ψ|2 is then given by

ρ(x, t) = ρ0

[1 − cos2β sech2

(√ρ0 cos

(β(x− V t− ξ0

))]. (10.195)

This is called a dark soliton because the density ρ(x, t) is minimized at the center ofthe soliton, where ρ = ρ0 sin

2β, which is smaller than the asymptotic |x| → ∞ value ofρ(±∞, t) = ρ0.

solitonssolitons starting in the 19th century, researchers found that certain nonlinear pdes admit...

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